Skip to main content Accessibility help
×
Home

Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains

  • Joseph Glaz (a1), Martin Kulldorff (a2), Vladimir Pozdnyakov (a1) and J. Michael Steele (a3)

Abstract

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Statistics, University of Connecticut, 215 Glenbrook Road, U-4120, Storrs, CT 06269-4120, USA.
∗∗ Postal address: Department of Ambulatory Care and Prevention, Harvard Medical School, 133 Brookline Avenue, Boston, MA 02215-3920, USA.
∗∗∗ Email address: vladimir.pozdnyakov@uconn.edu
∗∗∗∗ Postal address: Department of Statistics, Wharton School, Huntsman Hall 447, University of Pennsylvania, Philadelphia, PA 19104, USA.

References

Hide All
Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems and failure runs in higher order Markov dependent trials. Ann. Inst. Statist. Math. 48, 773787.
Antzoulakos, D. (2001). Waiting times for patterns in a sequence of multistate trials. J. Appl. Prob. 38, 508518.
Benevento, R. V. (1984). The occurrence of sequence patterns in ergodic Markov chains. Stoch. Process. Appl. 17, 369373.
Biggins, J. D. and Cannings, C. (1987a). Formulas for mean restriction-fragment lengths and related quantities. Amer. J. Hum. Genet. 41, 258265.
Biggins, J. D. and Cannings, C. (1987b). Markov renewal processes, counters and repeated sequences in Markov chains. Adv. Appl. Prob. 19, 521545.
Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518531.
Breen, S., Waterman, M. and Zhang, N. (1985). Renewal theory for several patterns. J. Appl. Prob. 22, 228234.
Chryssaphinou, O. and Papastavridis, S. (1990). The occurrence of a sequence of patterns in repeated dependent experiments. Theory Prob. Appl. 35, 145152.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n: F systems with (k-1)-step Markov dependence. IEEE Trans. Reliab. 35, 602606.
Fu, J. C. (2001). Distribution of the scan statistics for a sequence of bistate trials. J. Appl. Prob. 38, 908916.
Fu, J. C. and Chang, Y. (2002). On probability generating functions for waiting time distribution of compound patterns in a sequence of multistate trials. J. Appl. Prob. 39, 7080.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 78, 168175.
Gerber, H. and Li, S. (1981). The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Process. Appl. 11, 101108.
Guibas, L. and Odlyzko, A. (1981a). Periods of strings. J. Combinatorial Theory A 30, 1942.
Guibas, L. and Odlyzko, A. (1981b). String overlaps, pattern matching and nontransitive games. J. Combinatorial Theory A 30, 183208.
Han, Q. and Hirano, K. (2003). Sooner and later waiting time problems for patterns in Markov dependent trials. J. Appl. Prob. 40, 7386.
Li, S. (1980). A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 11711176.
Pozdnyakov, V. and Kulldorff, M. (2006). Waiting times for patterns and a method of gambling teams. Amer. Math. Monthly 113, 134143.
Pozdnyakov, V., Glaz, J., Kulldorff, M. and Steele, J. M. (2005). A martingale approach to scan statistics. Ann. Inst. Statist. Math. 57, 2137.
Robin, S. and Daudin, J.-J. (1999). Exact distribution of word occurrences in a random sequence of letters. J. Appl. Prob. 36, 179193.
Stefanov, V. T. (2000). On some waiting time problems. J. Appl. Prob. 37, 756764.
Stefanov, V. T. (2003). The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach. J. Appl. Prob. 40, 881892.
Stefanov, V. T. and Pakes, A. G. (1997). Explicit distributional results in pattern formation. Ann. Appl. Prob. 7, 666678.
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables. Ann. Inst. Statist. Math. 50, 655671.
Williams, D. (1991). Probability with Martingales. Cambridge University Press.

Keywords

MSC classification

Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains

  • Joseph Glaz (a1), Martin Kulldorff (a2), Vladimir Pozdnyakov (a1) and J. Michael Steele (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed